Find each root. See Example 3. -∛512

Verified step by step guidance

1. Recognize that the problem asks for the cube root of -512, which means finding a number \( x \) such that \( x^3 = -512 \).
2. Recall that the cube root function can be applied to negative numbers because \( (-a)^3 = -a^3 \). So, \( \sqrt[3]{-512} = -\sqrt[3]{512} \).
3. Focus on finding \( \sqrt[3]{512} \). Express 512 as a power of a smaller number, for example, check if 512 is a power of 2: \( 512 = 2^9 \).
4. Use the property of roots and exponents: \( \sqrt[3]{2^9} = 2^{9/3} = 2^3 \).
5. Calculate \( 2^3 \) to find the cube root of 512, then apply the negative sign to get the cube root of -512.

Key Concepts

Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27. Cube roots can be positive or negative depending on the original number.

Handling Negative Radicals

When taking the cube root of a negative number, the result is also negative because a negative number multiplied three times remains negative. For instance, the cube root of -512 is -8 since (-8)³ = -512. This differs from even roots, where negative inputs are not real.

Prime Factorization for Roots

Prime factorization breaks a number into its prime components, making it easier to find roots. For example, 512 = 2⁹, so the cube root of 512 is 2³ = 8. Using prime factorization simplifies root extraction by grouping factors according to the root's degree.